Degeneracy Breaking in Some Frustrated Magnets

1. Degeneracy Breaking in Some Frustrated Magnets Bangalore Mott Conference, July 2006

2. Outline • Motivation: Why study frustrated magnets? • Chromium spinels and magnetization plateau • Quantum dimer model and its phase diagram • Constrained phase transitions and exotic criticality

3. Degeneracy “breaking” • Origin of (most) magnetism: Hund’s rule • Splitting of degenerate atomic multiplet • Degeneracy “quenches” kinetic energy and makes interactions dominant • Degeneracy on a macroscopic scale • Macroscopic analog of “Hund’s rule” physics • Landau levels ) FQHE • Large-U Hubbard model ) High-Tc ? • Frustrated magnets) • Spin liquids ?? • Complex ordered states • Exotic phase transitions ??

4. + + … Spin Liquids? • Anderson: proposed RVB states of quantum antiferromagnets • Phenomenological theories predict such states have remarkable properties: • topological order • deconfined spinons • There are now many models exhibiting such states

5. + + Quantum Dimer Models … Rohksar, Kivelson Moessner, Sondhi Misguich et al • Models of “singlet pairs” fluctuating on lattice (can have spin liquid states) • Constraint: 1 dimer per site. • Construction problematic for real magnets • non-orthogonality • not so many spin-1/2 isotropic systems • dimer subspace projection not controlled • We will find an alternative realization, more akin to spin ice

6. cubic Fd3m Chromium Spinels Takagi group ACr2O4 (A=Zn,Cd,Hg) • spin S=3/2 • no orbital degeneracy • isotropic • Spins form pyrochlore lattice • Antiferromagnetic interactions CW = -390K,-70K,-32K for A=Zn,Cd,Hg

7. Pyrochlore Antiferromagnets • Heisenberg • Many degenerate classical configurations • Zero field experiments (neutron scattering) • Different ordered states in ZnCr2O4, CdCr2O4 • HgCr2O4? • What determines ordering not understood c.f. CW = -390K,-70K,-32K for A=Zn,Cd,Hg

8. Magnetization Process H. Ueda et al, 2005 • Magnetically isotropic • Low field ordered state complicated, material dependent • Plateau at half saturation magnetization in 3 materials

9. HgCr2O4 neutrons • Neutron scattering can be performed on plateau because of relatively low fields in this material. H. Ueda et al, unpublished • Powder data on plateau indicates quadrupled (simple cubic) unit cell with P4332 space group • We will try to understand this ordering

10. Collinear Spins • Half-polarization = 3 up, 1 down spin? • - Presence of plateau indicates no transverse order • Spin-phonon coupling? • - classical Einstein model large magnetostriction Penc et al H. Ueda et al effective biquadratic exchange favors collinear states But no definite order • “Order by disorder” • in the semiclassical S!1 limit, thermal and quantum fluctuations favor collinear states (Henley…) • this alone probably gives a rather narrow plateau if at all • the S= theory does not fully resolve the degeneracy

11. 3:1 States • Set of 3:1 states has thermodynamic entropy • - Less degenerate than zero field but still degenerate • - Maps to dimer coverings of diamond lattice • Effective dimer model: What splits the degeneracy? • Classical: • further neighbor interactions • Lattice coupling beyond Penc et al? • Quantum?

12. Ising Expansion following Hermele, M.P.A. Fisher, LB • Strong magnetic field breaks SU(2) ! U(1) • Substantial polarization: Si? < Siz • Formal expansion in J?/Jz reasonable (carry to high order) 3:1 GSs for 1.5<h<4.5 • Obtain effective hamiltonian by DPT in 3:1 subspace • First off-diagonal term at 9th order! [(6S)th order] • First non-trivial diagonal term at 6th order! • Techniques can be applied to any lattice of corner-sharing simplexes • - kagome, checkerboard… = Off-diagonal:

13. + + Effective Hamiltonian Diagonal term: State Dominant? • Checks: • Two independent techniques to sum 6th order DPT • Agrees exactly with large-s calculation (Hizi+Henley) in overlapping limit and resolves degeneracy at O(1/s) Extrapolated V ¼ -2.3K

14. + + Quantum Dimer Model on diamond lattice • Expected phase diagram (various arguments) Maximally “resonatable” R state U(1) spin liquid “frozen” state 1 S=1 -2.3 0 Rokhsar-Kivelson Point • Interesting phase transition between R state and spin liquid! Will return to this. Quantum dimer model is expected to yield the R state structure Caveat: other diagonal terms can modify phase diagram for large negative v

15. R state • Unique state saturating upper bound on density of resonatable hexagons • Quadrupled (simple cubic) unit cell • Still cubic: P4332 • 8-fold degenerate • Quantum dimer model predicts this state uniquely.

16. Is this the physics of HgCr2O4? • Probably not: • Quantum ordering scale » |V| » 0.02J • Actual order observed at T & Tplateau/2 • We should reconsider classical degeneracy breaking by • Further neighbor couplings • Spin-lattice interactions • C.f. “spin Jahn-Teller”: Tchernyshyov et al Considered identical distortions of each tetrahedral “molecule”

17. Einstein Model vector from i to j • Site phonon • Optimal distortion: • Lowest energy state maximizes u*: • “bending rule”

18. Bending Rule States • At 1/2 magnetization, only the R state satisfies the bending rule globally • - Einstein model predicts R state! • Zero field classical spin-lattice ground states? Unit cells states • collinear states with bending rule satisfied for both polarizations • ground state remains degenerate • Consistent with different zero field ground states for A=Zn,Cd,Hg • Simplest “bending rule” state (weakly perturbed by DM) appears to be consistent with CdCr2O4 Chern et al, cond-mat/0606039

19. Constrained Phase Transitions • Schematic phase diagram: T Magnetization plateau develops T» J R state Classical (thermal) phase transition Classical spin liquid “frozen” state 1 0 U(1) spin liquid • Local constraint changes the nature of the “paramagnetic” state • - Youngblood+Axe (81): dipolar correlations in “ice-like” models • Landau-theory assumes paramagnetic state is disordered • - Local constraint in many models implies non-Landau classical criticality Bergman et al, PRB 2006

20. Dimer model = gauge theory • Can consistently assign direction to dimers pointing from A ! B on any bipartite lattice B A • Dimer constraint ) Gauss’ Law • Spin fluctuations, like polarization fluctuations in a dielectric, have power-law dipolar form reflecting charge conservation

21. A simple constrained classical critical point • Classical cubic dimer model • Hamiltonian • Model has unique ground state – no symmetry breaking. • Nevertheless there is a continuous phase transition! • - Analogous to SC-N transition at which magnetic fluctuations are quenched (Meissner effect) • - Without constraint there is only a crossover.

22. Numerics (courtesy S. Trebst) C Specific heat T/V “Crossings”

23. Conclusions • Quantum and classical dimer models can be realized in some frustrated magnets • This effective model can be systematically derived by degenerate perturbation theory • Rather general methods can be applied to numerous problems • Spin-lattice coupling probably is dominant in HgCr2O4, and a simple Einstein model predicts a unique and definite state (R state), consistent with experiment • Probably spin-lattice coupling plays a key role in numerous other chromium spinels of current interest (multiferroics). • Local constraints can lead to exotic critical behavior even at classical thermal phase transitions. • Experimental realization needed!